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3.1: One-Step Equations

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Learning Objectives

At the end of this lesson, students will be able to:

  • Solve an equation using addition.
  • Solve an equation using subtraction.
  • Solve an equation using multiplication.
  • Solve an equation using division.

Vocabulary

Terms introduced in this lesson:

equal
equation
isolate
linear equation

Teaching Strategies and Tips

Use the introductory problem to motivate equivalent equations, since it can be solved two ways:

  • The cost plus the change received is equal to the amount paid, x + 22 = 100.
  • The cost is equal to the difference between the amount paid and the change, x = 100 - 22

Examples 1 and 2 are essentially the same; constant terms must be added to both sides to isolate x on the left. Adding vertically can benefit students.

Additional Example.

Solve 12 = -4 + x.

Solution. To isolate x, add 4 to both sides of the equation. Add vertically.

12 & = -\cancel{4} + x\\+4 & = +\cancel{4}\\16 & = x

The variable in Example 3 is not the usual x. Remind students that the letter of the variable does not matter. Additional Example.

Solve -21=n+14.

Hint: To isolate n, subtract 14 from both sides of the equation.

In Examples 4-6, teachers may opt to solve the equations by adding the opposite in lieu of subtracting.

Example.

Solve -17=x+8.

Hint: To isolate x, add -8 to both sides of the equation. Add vertically.

-17 & = x+\cancel{8}\\-8 & = -\cancel{8}\\-25 & = x

Use Example 6 as an example of an equation with fractions. Remind students to find common denominators.

Point out in Example 8 that in general,

\frac{ax}{b}=\frac{a}{b}x

which will help students isolate x in one step, multiplying by the reciprocal of a/b.

Note that the equation in Example 10 can be written in dollars or in cents:

  • 5x=3.25 (x in dollars)
  • 5x=325 (x in cents)

Although Examples 13 and 15 can be solved by making a table and Example 14 by guessing and checking, teachers are encouraged to help students setup and solve an equation of the type presented in the lesson.

Error Troubleshooting

General Tip: After the constant term is canceled in a one-step equation, the variable must be carried down onto the next line. Remind students to write the x =.

General Tip: Students forget to perform the same operation on both sides of an equation. Have students use a colored pencil to write what they are doing to both sides of the equation.

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